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Full text of "Mathematical And Physical Papers - Iii"

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mass of the fluid displaced by the sphere, which is equal to f 7r/oa3,
we get
F= - M'cn whence
Since /s/— 1 has been eliminated, this equation will remain unchanged when we pass from the symbolical to the real values of .Fand f
Let T be the time of oscillation from rest to rest, so that UT—TT, and put for shortness &, Jc' for the coefficients of Mr in (51); then
y=s   /JL,   & = J+A     &' = A-(l + 1>) ...... (52).
V 2ft V           ^    4m           4z/a V       vaj         ^
The first term in the expression for the force F has the same effect as increasing the inertia of the sphere. To take account of this term, it will be sufficient to conceive a mass JcM ' collected at the centre of the sphere, adding to its inertia without adding to its weight. The main effect of the second term is to produce a diminution in the arc of oscillation : its effect on the time of oscillation would usually be quite insensible, and must in fact be neglected for consistency's sake, because the motion of the fluid was determined by supposing the motion of the sphere permanent, which is only allowable when we neglect the square of the rate of decrease of the arc of oscillation.
If we form the equation of motion of the sphere, introducing the force F, and then proceed to integrate the equation, we shall obtain in the integral an exponential €~8* multiplying the circular function, 8 being half the coefficient of dg/dt divided by that of d^/df. Let M be the mass of the sphere, My* its moment of inertia about the axis of suspension, then
nk'M' (I + a)2 = 28 {M<f + kM' (I + a)2}.
In considering the diminution of the arc of oscillation, we may put  I -f a for 7.    During i oscillations, let the arc of oscillation be diminished in the ratio of A0 to Ait then ,      A.     . .     iri    Jc'M1